Applying an iterative method numerically to solve n × n rsta.royalsocietypublishing.org matrix Wiener–Hopf equations with exponential factors Research Matthew J. Priddin1, Anastasia V. Kisil2 Article submitted to journal and Lorna J. Ayton1 1DAMTP, University of Cambridge, Cambridge, CB3 Subject Areas: 0WA, UK 2Department of Mathematics, The University wave motion, applied mathematics, of Manchester, Manchester, M13 9PL, UK differential equations Keywords: This paper presents a generalisation of a recent iterative approach to solving a class of 2 2 Wiener–Hopf equations, × matrix Wiener–Hopf equations involving exponential Riemann-Hilbert problem, iterative factors. We extend the method to square matrices of methods, scattering, n-part arbitrary dimension n, as arise in mixed boundary boundaries value problems with n junctions. To demonstrate the method we consider the classical problem of Author for correspondence: scattering a plane wave by a set of collinear plates. The results are compared to other known Matthew Priddin methods. We describe an effective implementation e-mail: [email protected] using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy. 1. Introduction The Wiener–Hopf technique [1] provides a powerful tool to approach varied problems such as the solution of integral equations and random processes. Applications may so be found in such diverse fields as aeroacoustics [2], metamaterials [3], geophysics [4], crack propagation [5] and financial mathematics [6]. The mathematical problem underlying the Wiener– Hopf technique may be simply stated: find functions that are analytic in adjacent (or overlapping) regions of a complex manifold that satisfy a prescribed jump condition at their common boundary. c The Author(s) Published by the Royal Society. All rights reserved. α Often, as in this work, the manifold is the complex plane C, and the regions are the upper 2 + 2 and lower half-planes, respectively = (α) > c and − = (α) < c where c is an arbitrary D = − D = rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000 small constant. The jump between two unknown vector functions is described by an equation .................................................................. K(α)Ψ (α) + Ψ+(α) = F (α) (1.1) − valid on some strip c < (α) < c, where K(α) is a known matrix function and + and − = − decorations denote analyticity in the upper and lower half-planes respectively. The classical Wiener–Hopf technique exploits the analytic structure of the problem by + requiring a special multiplicative factorisation of the kernel K into two parts K = K−K , each analytic and zero-free in one of the adjacent regions. Such a factorisation has been long known to exist, but generic, stable and constructive approaches are unknown [7]. The stability and well- posedness of the problem rests on certain invariants of the matrix kernel K, the partial indices, though the only general method to determine these is to solve the hard problem of finding a factorisation [8,9]. We note that in this paper we do not find the matrix factorisation and hence do not determine the partial indices (we know of no results of how to find the partial indices for our class of matrix Wiener-Hopf problems). However, since the class of matrix functions for which our method applies results from well-posed physical problems we believe it should be stable under small perturbations. These difficulties surrounding matrix factorisation mean that in practice it is typical to resort to approximate methods: approximating the matrix, K, by one more easily factored, either exploiting a small parameter perhaps associated with a particular physical regime [10,11], or a suitable approximation space as in Padé approximation [12]. One might also undertake the factorisation process by ‘singularity removal’: introducing functions defined to additively remove the singularities, solving the Wiener–Hopf equation and if necessary subsequently determining any unknown parameters of these functions by enforcing this removal, perhaps approximately [13,14]. When solving mixed boundary value problems using the Wiener–Hopf technique the number of unknown functions is associated with the number of distinct boundary sections on which boundary values must be found. Therefore the solution of matrix problems is vital to exploit the Wiener–Hopf technique to solve physical problems involving multiple changes in boundary condition. Despite its importance, as discussed above, there are sparse methods available to solve general matrix Wiener–Hopf problems [8,9,15]. In addition, much of the existing literature is concerned with the factorisation of 2 2 matrices, leaving the more general and challenging × case of n n matrix functions relatively unexplored [16–18] (especially where n > 4). Exponential × factors in the matrix kernel, often associated with the displacement between boundary junctions, pose particular difficulty in part due to their non-factorability. Previous works have considered the restricted cases of 2 2 matrix Wiener–Hopf problems with exponential factors [19,20] and × n n Riemann–Hilbert problems with meromorphic coefficients [21]. A numerical Riemann– × Hilbert formulation [22] has been recently adapted to solve classical Wiener–Hopf problems [23] and certain problems involving exponential factors [24], each relying on the choice of a suitable basis. The numerous potential applications of n n Wiener–Hopf problems involving × exponential factors underlines the importance of developing suitable methods [25,26]. This paper helps to fill this gap by extending the use of the iterative Wiener–Hopf method introduced in [20] for the two dimensional matrix problem to n dimensions. This method has the particular attraction that each step may permit a physically meaningful interpretation. The two dimensional method has found applications in aeroacoustic [27] and crack propagation problems [28]. In the former, aeroacoustic scattering from a finite porous extension to a rigid trailing edge, each Wiener–Hopf factorisation was undertaken by brute force. In the latter, a crack propagation problem built on the application of the Wiener–Hopf technique in fracture mechanics pioneered in numerous works by Slepyan[29,30]. Importantly, by implementing an effective numerical factorisation procedure based on techniques presented in [22,31–33] this paper shows that this method provides a practical and fast procedure for solving problems involving larger n n matrices. × ik(x cos θi+y sin θi) incident wave φi(x, y) = e− scattered wave φ(x, y) 3 rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 0000000 .................................................................. θi ··· x0 = x1 x2 x3 x4 x2N 1 x2N x = + −∞ − 2N+1 ∞ (a) set of collinear finite plates ··· x0 = x1 x2 x3 x4 x2N 1 x = + −∞ − 2N ∞ (b) set of collinear finite plates and one semi-infinite plate ··· x = x1 x2 x3 x2N x = + 0 −∞ 2N+1 ∞ (c) set of apertures in an infinite plate (collinear finite plates and two semi-infinite plates) Figure 1: Scattering of a plane wave by a set of collinear plates. The set of plates may extend a finite length in (a) none, (b) one or (c) both of the positive and negative x directions. This new numerical approach is readily applicable to scattering by a set of collinear plates, such as the case of several finite plates shown in figure 1a. This class of geometries encompasses a number of practical and canonical problems, and will be the focus of this paper to demonstrate the iterative method. The Sommerfeld diffraction problem of scattering by a half-plane presents the prototypical scattering model. It is the classical example to introduce the Wiener–Hopf technique [1], dealing with both an unbounded domain and the sharp material junction. The diffraction of a wave by one finite plate (two junctions) has been extensively studied [34–36], but the case of two plates is much harder. Scattering by more than two plates has been explicitly considered using Wiener–Hopf but neglecting edge interactions in [37,38], analytically generalising special functions [39] and numerically in [33,40,41]. Another interesting rigorous approach is to reduce the n-plate problem to the solution of a simple ordinary differential equation [42], here the practical difficulty lies in finding some constant parameters of this ODE. A common theme in many approaches is the successive treatment of diffraction events to yield Schwarzschild diffraction series; our implementation may be interpreted as recovering a diffraction series solution in the spectral domain (Fourier space). The fact that the proposed method is devised in the spectral domain rather than in the physical space is key. This allows to take advantage of the singularity structures of the kernel in the complex plane and hence perform hundreds rather than one or two iterations, see Section3(c). Significantly, the proposed method applies to some Wiener-Hopf problems arising in crack propagation or Lévi processes, which are not related to diffraction and where a Schwarzschild diffraction series approach would not be a natural choice. General numerical boundary based methods, that avoid the need for Wiener–Hopf factorisation, can provide effective solutions to scattering problems from convex sets of objects, though typically these require a linear growth in